The first half of the statement is exactly [1]} restricted to \(0,1\) -matrices.
Although it is not stated, the proof of [1]} also verifies
the additional requirement that only one row or column is extracted from each part.
We only need to carefully analyse it.
First, [1]} shows that matrices with rank-\(f_1(k)\)  divisions
have submatrices with rank-\(k\)  Latin divisions—a somewhat technical variant of rank divisions.
Crucially, the proof constructs a Latin division from a high-rank division by
deleting rows or columns, and merging parts of the division.
Then, [1]} extracts some encoding of permutations from these Latin divisions
by deleting additional rows and columns, and finally picking a single row, resp. column, by part.
Finally, [1]} shows that these encodings of permutations
can be refined to be one of \(\mathcal {M}_=,\mathcal {M}_{\ne },\mathcal {M}_{\leqslant _R},\mathcal {M}_{\geqslant _R},\mathcal {M}_{\leqslant _C},\mathcal {M}_{\geqslant _C}\) .
To summarize, the proof of [1]} consists only of
extractions of submatrices, and merging of parts of the division,
with one step at which a single row or column is picked per part of the division.
This ensures the desired property.
